Homomorphism of independent random variable convolution and matrix multiplication

TL;DR

This paper demonstrates a homomorphism between the convolution of independent random variables over finite groups and matrix multiplication of doubly stochastic matrices, providing insights into the limiting distributions of certain stochastic processes.

Contribution

It establishes a homomorphic relationship between convolutions of independent variables and matrix multiplication, offering a new perspective and a simplified proof for the uniformity of limiting distributions.

Findings

Convolution over finite groups corresponds to matrix multiplication of doubly stochastic matrices.

Limiting distributions of stationary independent increment processes are always uniform.

Provides a short, elegant proof of a classical theorem.

Abstract

A map is given showing that convolutions of independent random variables over a finite group and matrix multiplications of doubly stochastic matrices are homomorphic. As an application, a short proof is given to the theorem that the limiting distributions of stochastic processes with stationary independent increments over a finite group are always uniform.